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Geometry has long been studied and considered sacred for its ability to represent and make comprehensible the myriad phenomena of the natural world. Geometry is idealized form, understood in the mind and can be represented in the two-dimensional realm of drawing. Other geometries exist outside of our minds, in the physical world and can demonstrate universal truths and orders which cause us to be. The act of geometrical demonstration allows for the construction of invisible forces, orders, and patterns underpinning the physical world. This work consists of a series of primarily perspectival drawings which rely on the idea of proportion as a means to construct, demonstrate, and represent potential architectures. / Master of Architecture / Things, such as architecture, should be in accord with the profound order of the world. That order can be revealed with geometry, which governs proportion, and is traced in this work to investigate universal orders via constructed drawings, especially perspectives.

Play is important in a child's development, growth and education. Children must be given a space where--in place of formal education--wonder and the love of play can be fostered and encouraged, allowing them to transition into becoming young individuals. By constructing a building with them in mind, children are offered opportunities to discover, play, and wonder. / Master of Architecture / This thesis challenges the typical symmetrical, standard, or traditional school system in an attempt to teach children concepts of light, shadow and color, geometry and to provide them with a sense of their natural environment, or surroundings all while inviting them to learn in a playful matter.

Yes / In this paper we describe a novel concept for classification of complex parametric geometry based on the concept of the Divider Set. The Divider Set is an alternative concept to maximal disks, Voronoi sets and cut loci. The Divider Set is based on a formal definition relating to topology and differential geometry. In this paper firstly we discuss the formal definition of the Divider Set for complex 3-dimensional geometry. This is then followed by the introduction of a computationally feasible algorithm for computing the Divider Set for geometry which can be defined in explicit parametric form. Thus, an explicit solution form taking advantage of the special form of the parametric geometry is presented. We also show how the Divider Set can be computed for various complex parametric geometry by means of illustrating our concept through a number of examples

In recent decades, education researchers have recognized the need for teachers to have a nuanced content knowledge in addition to pedagogical knowledge, but very little research was conducted into what this knowledge would entail. Beginning in 2008, math education researchers began to develop a theoretical framework for the mathematical knowledge needed for teaching, but their work focused primarily on elementary schools. I will present an analysis of the mathematical knowledge needed for teaching about the regular curves and surfaces, two important concepts in differential geometry which generalize to the advanced notion of a manifold, both in a college classroom and in an on-line format. I will also comment on the philosophical and political questions that arise in this analysis.

The cohomology of moduli spaces of curves has been extensively studied in classical algebraic geometry. The emergent field of tropical geometry gives new views and combinatorial tools for treating these classical problems. In particular, we study the cohomology of heavy/light Hassett spaces, moduli spaces of heavy/light weighted stable curves, denoted as $\calm_{g, w}$ for a particular genus $g$ and a weight vector $w \in (0, 1]^n$ using tropical geometry. We survey and build on the work of \citet{Cavalieri2014}, which proved that tropical compactification is a \textit{wonderful} compactification of the complement of hyperplane arrangement for these heavy/light Hassett spaces. For $g = 0$, we want to find the tropicalization of $\calm_{0, w}$, a polyhedral complex parametrizing leaf-labeled metric trees that can be thought of as Bergman fan, which furthermore creates a toric variety $X_{\Sigma}$. We use the presentation of $\overline{\calm}_{0,w}$ as a tropical compactification associated to an explicit Bergman fan, to give a concrete presentation of the cohomology.

This thesis is mainly concerned with the study of hypersymplectic structures in gauge theory. These structures arise via applications of the hypersymplectic quotient construction to the action of the gauge group on certain spaces of connections and Higgs fields. Motivated by Kobayashi-Hitchin correspondences in the case of hyperkähler moduli spaces, we first study the relationship between hypersymplectic, complex and paracomplex quotients in the spirit of Kirwan's work relating Kähler quotients to GIT quotients. We then study dimensional reductions of the ASD equations on $mathbb R^{2,2}$. We discuss a version of twistor theory for hypersymplectic manifolds, which we use to put the ASD equations into Lax form. Next, we study Schmid's equations from the viewpoint of hypersymplectic quotients and examine the local product structure of the moduli space. Then we turn towards the integrability aspects of this system. We deduce various properties of the spectral curve associated to a solution and provide explicit solutions with cyclic symmetry. Hitchin's harmonic map equations are the split signature analogue of the self-duality equations on a Riemann surface, in which case it is known that there is a smooth hyperkähler moduli space. In the case at hand, we cannot expect to obtain a globally well-behaved moduli space. However, we are able to construct a smooth open set of solutions with small Higgs field, on which we then analyse the hypersymplectic geometry. In particular, we exhibit the local product structures and the family of complex structures. This is done by interpreting the equations as describing certain geodesics on the moduli space of unitary connections. Using this picture we relate the degeneracy locus to geodesics with conjugate endpoints. Finally, we present a split signature version of the ADHM construction for so-called split signature instantons on $S^2 imes S^2$, which can be given an interpretation as a hypersymplectic quotient.